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KoszulDivisorOnPic14M8 :: randomCanonicalCurveGenus8with8Points

randomCanonicalCurveGenus8with8Points -- compute a random canonical curve of genus 8 with 8 marked point

Synopsis

Description

According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the transversal intersection C=ℙ7 ∩ G(2,6) ⊂ ℙ15. In particular this is true for the general curve of genus 8. Picking 8 points in the Grassmannian G(2,6) at random and ℙ7 as their span gives the result.

i1 : FF=ZZ/10007;S=FF[x_0..x_7];
i3 : (I,points)=randomCanonicalCurveGenus8with8Points S;
i4 : betti res I

            0  1  2  3  4  5 6
o4 = total: 1 15 35 42 35 15 1
         0: 1  .  .  .  .  . .
         1: . 15 35 21  .  . .
         2: .  .  . 21 35 15 .
         3: .  .  .  .  .  . 1

o4 : BettiTally
i5 : points

o5 = {ideal (x  + 2218x , x  - 4540x , x  - 626x , x  - 4294x , x  + 1276x ,
              6        7   5        7   4       7   3        7   2        7 
     ------------------------------------------------------------------------
     x  - 2638x , x  - 2532x ), ideal (x  + 3421x , x  - 3522x , x  - 3180x ,
      1        7   0        7           6        7   5        7   4        7 
     ------------------------------------------------------------------------
     x  + 1692x , x  + 1898x , x  - 276x , x  - 2805x ), ideal (x  + 2537x ,
      3        7   2        7   1       7   0        7           6        7 
     ------------------------------------------------------------------------
     x  + 3976x , x  - 4098x , x  + 4833x , x  + 2403x , x  + 3542x , x  -
      5        7   4        7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     2455x ), ideal (x  + 4468x , x  - 922x , x  - 613x , x  + 4475x , x  -
          7           6        7   5       7   4       7   3        7   2  
     ------------------------------------------------------------------------
     448x , x  - 972x , x  + 4770x ), ideal (x  - 4035x , x  - 1606x , x  -
         7   1       7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     3277x , x  + 2598x , x  + 3081x , x  + 4219x , x  - 3350x ), ideal (x  +
          7   3        7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     4479x , x  - 467x , x  + 2935x , x  + 885x , x  + 1579x , x  - 548x , x 
          7   5       7   4        7   3       7   2        7   1       7   0
     ------------------------------------------------------------------------
     + 126x ), ideal (x  + 4996x , x  - 3771x , x  - 4847x , x  - 4789x , x 
           7           6        7   5        7   4        7   3        7   2
     ------------------------------------------------------------------------
     + 648x , x  - 4446x , x  - 4463x ), ideal (x  - 3955x , x  + 2510x , x 
           7   1        7   0        7           6        7   5        7   4
     ------------------------------------------------------------------------
     - 3643x , x  + 2825x , x  + 1886x , x  - 2116x , x  - 2821x )}
            7   3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)